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AdataenvelopmentanalysismodelwithintervaldataandundesirableoutputforcombinedcyclepowerplantperformanceassessmentKavehKhalili-Damghania,MadjidTavanab,c,,ElhamHaji-SaamiaaDepartmentofIndustrialEngineering,South-TehranBranch,IslamicAzadUniversity,Tehran,IranbBusinessSystemsandAnalyticsDepartment,LindbackDistinguishedChairofInformationSystemsandDecisionSciences,LaSalleUniversity,Philadelphia,PA19141,USAcBusinessInformationSystemsDepartment,FacultyofBusinessAdministrationandEconomics,UniversityofPaderborn,D-33098Paderborn,GermanyarticleinfoArticlehistory:Availableonline30August2014Keywords:EconomicscalesizeReturntoscaleDataenvelopmentanalysisImprecisedata:undesirableoutputsCombinedcyclepowerplantabstractDeterminingtheoptimalscalesizeofacombinedcyclepowerplantisinherentlyacomplexproblemoftenwithmultipleandconictingcriteriaaswellasuncertainfactors.
Thecomplexityoftheproblemiscompoundedbytheproductionofundesirableoutputsandthepresenceofnaturalandmanagerialdisposability.
Weproposeacustomizeddataenvelopmentanalysis(DEA)methodforsolvingthereturntoscale(RS)probleminthepresenceofuncertaindataandundesirableoutputs.
Acombinedcyclepowerplantisconsideredadecisionmakingunit(DMU)whichconsumesfuelstoproduceelectricityandemis-sions.
Theuncertaintyoftheinputsandoutputsaremodeledwithintervaldataandtheemissionsareassumedtobeundesirableoutputs.
TheproposedDEAmethoddeterminestheintervalefciencyscoresoftheDMUsandoffersapracticalbenchmarkforenhancingtheefciencyscores.
Wedemonstratetheapplicabilityoftheproposedmethodandexhibittheefcacyoftheprocedurewithasix-yearstudyof17combinedcyclepowerplantsinIran.
Themaincontributionsofthispaperaresixfold:we(1)modeltheuncertaintiesintheinputandoutputdatausingintervaldata;(2)considerundesirableoutputs;(3)determinetheefciencyscoresoftheDMUsasintervalvalues;(4)developagroupofindicestodistinguishbetweentheefcientandinefcientDMUs;(5)determinethemosteconomicscalesizefortheefcientDMUs;and(6)determinepracticalbenchmarksfortheinefcientDMUs.
2014ElsevierLtd.
Allrightsreserved.
1.
IntroductionEnergygeneration,distributionandconsumptionplayavitalroleintheeconomy.
Generationanddistributionofcleanenergyandsustainabledevelopmentconsiderationsareessentialforfuturegenerationsespeciallyindevelopingcountries.
Therefore,energysystemsshouldsystematicallybeassessedandupdatedinordertoimprovetheperformance,todeterminetheeconomicscalesizeandtoefcientlyutilizeresources.
Agreatdealofresearchhasbeendevotedtotheeldofenergysystemassessmentandevaluationduringrecentyears.
Azade,Ghaderi,andMaghsoudi(2008)proposedanintegratedhierarchi-calapproachbasedondataenvelopmentanalysis(DEA),principalcomponentanalysisandnumericaltaxonomytolocatesolarplantsindifferentregionsandcitiesinIran.
Aguirre,Villalobos,Phelan,andPacheco(2011)presentedamethodologytomeasurerelativeindustrialenergyefciencyacrossplantswithinamanufacturingsectorthroughtheuseofenergy-productionsignatures.
Theyusedlinearprogramming,regression,benchmarkingandsimulationmodelstostudythebehaviorofarepresentativemanufacturingplant.
ThemethodologywasvalidatedusingdatafromtheDepart-mentofEnergydatabase.
Bampatsou,Papadopoulos,andZervas(2013)usedaDEAmodeltodeterminethetechnicalefciencyindexofEU-15countries.
Salazar-Ordóez,Pérez-Hernández,andMartín-Lozano(2013)estimatedthepotentialefciencyofthesugarbeetcropinSpainusingaDEAmodel.
Kagawa,Takezono,Suh,andKudoh(2013)evaluatedtheefciencyofanadvancedbio-dieselplantinJapanusingDEA.
Fang,Hu,andLou(2013)usedaninput-orientedvariablereturntoscale(VRS)DEAmodeltocomputethepuretechnicalefciencyandenergy-savingtargets.
Fangetal.
(2013)utilizedafour-stageDEAandstudiedtheeffectsofindustrycharacteristicsontheenergy-savingtargets.
Chang,Zhang,Danao,andZhang(2013)proposedanon-radialDEAmodelwithslack-basedmeasurestoanalyzetheenvironmentalhttp://dx.
doi.
org/10.
1016/j.
eswa.
2014.
08.
0280957-4174/2014ElsevierLtd.
Allrightsreserved.
Correspondingauthorat:BusinessSystemsandAnalyticsDepartment,LindbackDistinguishedChairofInformationSystemsandDecisionSciences,LaSalleUniversity,Philadelphia,PA19141,USA.
Tel.
:+12159511129;fax:+12672952854.
E-mailaddresses:kaveh.
khalili@gmail.
com(K.
Khalili-Damghani),tavana@lasalle.
edu(M.
Tavana),elham.
hajisami@gmail.
com(E.
Haji-Saami).
URLs:http://kaveh-khalili.
webs.
com(K.
Khalili-Damghani),http://tavana.
us/(M.
Tavana).
ExpertSystemswithApplications42(2015)760–773ContentslistsavailableatScienceDirectExpertSystemswithApplicationsjournalhomepage:www.
elsevier.
com/locate/eswaefciencyoftheChinesetransportationsector.
Zhou,Xing,Fang,Liang,andXu(2013)proposedanewnon-radialDEAapproachbyintegratingtheentropyweightandtheslack-basedmodeltoevaluateefciencyoftheChinesepowerindustryattheprovinciallevel.
elen(2013)usedstochasticfrontieranalysistoanalyzetheefciencyperformancesofTurkishelectricitydistributioncompa-nies.
elen(2013)measured''howtheefciencyperformancesoftheelectricitydistributionregionswereaffectedbythemergesbetweendistributionregions''.
Kuosmanen,Saastamoinen,andSipilinen(2013)comparedtheimpactofthreemethodsoncostef-ciencyestimationandanalysis.
Kuosmanenetal.
(2013)validatedtheperformanceoftheirapproachusingMonteCarlosimulationsinastudyoftheelectricitydistributionindustryinFinland.
VazhayilandBalasubramanian(2013)proposeddeterministicandstochasticDEAmodelstooptimizeenergyplanningintheIndianpowersector.
Wu,An,Xiong,andChen(2013)proposedamethodthatcon-sideredundesirableoutputsandanalyzedcongestionofindustrialregionsinChina.
Bian,He,andXu(2013)evaluatedregionalenergyefciencyinChinabasedonanon-radialDEAmodel.
Bianetal.
(2013)treatednon-fossilenergyasaxedinputinordertomea-sureenergysavingsaswellasreduceCarbondioxide(CO2)emis-sionforimprovingefciency.
Zhang,Zhou,andChoi(2013)modeledenergyandCO2emissionperformanceinelectricitygen-erationandproposedameta-frontiernon-radialdirectionaldis-tancefunctiontoconsidertheheterogeneousgroupofelectricitygeneration,non-radialslacks,andundesirableoutputs,simulta-neously.
Zhangetal.
(2013)studiedelectricitygenerationinKoreaandestimatedtheCO2emissionsandthepotentialreductionsinenergyusageunderdifferenttechnologicalassumptions.
Riccardi,Oggioni,andToninelli(2012)estimatedtheefciencyofhighener-geticandCO2emissionsinthecementproductionprocessin21countries.
Riccardietal.
(2012)comparedstandardDEAmodelswithadirectionaldistancefunctionapproachtomeasuretheef-ciencyinthepresenceandintheabsenceofCO2emission.
SueyoshiandGoto(2010)reformulatedtheoriginalnon-radialDEAmodeltohandleundesirable(bad)outputs.
Theyappliedtheirmethodontheoperational,environmentalandcombinedef-ciencymeasuresofUScoal-redpowerplants.
Wu,Fan,Zhou,andZhou(2012)measuredindustrialenergyefciencybycon-structingbothstaticanddynamicenergyefciencyperformanceindices.
Wuetal.
(2012)consideredundesirableoutputssuchasCO2emissionsintheirmodelingframework.
Mandal(2010)evalu-atedtheIndiancementindustryinthepresenceofemissionsandundesirableoutputsusingaDEAmodel.
Althoughelectricityisconsideredaformofcleanenergy,pro-ductionofelectricityusinggas,steam,andcombinedcyclepowerplantsoftencausessomeemissionsandpollutions.
Therefore,theoutputofelectricityproductionisoftenmixedwithsomeunwantedandundesirableoutputs.
Consequently,itiscriticaltoimprovetheprocessofefciencymeasurementandthedetermina-tionofeconomicscalesizeofelectricitypowerplants.
Moreover,betterestimateofemissionsandpollutionsproducedbypowerplantscanenhancethestrategicplanningofsustainabledevelopment.
Theproblemofperformanceassessmentofelectricalpowerplantsischallengingandcomplex.
Thisproblemusuallyinvolvesmultipleandoftenconictingcriteriaandundesirableoutputswhichareoftendifculttoassessbecauseofenvironmentaluncer-tainties.
Theproductionofavarietyofemissions,pollutions,andotherundesirableoutputssuchasCO2causesfurthercomplications.
Tothebestofourknowledge,thereisnosinglemethodintheliteraturethatcanmeasuretheefciencyscoresandtodeterminethemosteconomicscalesizeofdecisionmakingunits(DMUs)inthepresenceofundesirableoutputsanduncertaindata.
Inthispaper,weproposeaDEAmethodformeasuringtheperformanceofcombinedcyclepowerplantsinthepresenceofpollutionpro-ductionanddatauncertainty.
AcombinedcyclepowerplantisassumedtobeaDMUwhichconsumesfossilfuelstoproduceelec-tricityasdesirableoutputsandpollutinggases(i.
e.
,CO2,SOxandNOx)asundesirableoutputs.
Moreover,theuncertaintyofinputsandoutputsduringtheplanninghorizonismodeledusingintervaldata.
Theproposedapproachisusedtodeterminethemosteconomicscalesizeofpowerplantsandtopresentpracticalsug-gestionsforefciencyimprovementofinefcientDMUs.
Thepro-posedmethodiscustomizedandappliedtoacasestudyofIranianelectricalpowerplantstoillustrateitsapplicabilityandefcacy.
Thetheoreticalcontributionsofthispaperaresixfold:we(1)modeltheuncertaintiesintheinputandoutputdatausingintervaldata;(2)considerundesirableoutputs;(3)determinetheefciencyscoresoftheDMUsasintervalvalues;(4)developagroupofindicestodistinguishbetweentheefcientandinefcientDMUs;(5)determinethemosteconomicscalesizefortheefcientDMUs;and(6)determinepracticalbenchmarksfortheinefcientDMUs.
Theremainderofthispaperisorganizedasfollows.
InSection2,webrieyreviewtherelevantliteratureontheconventionalDEAmodels,uncertaintyinDEAmodels,undesirableoutputsinDEAmodels,RSinDEAmodels,andtheapplicationsofDEAmodelsinpowerplantsassessmentandenergygeneration.
InSection3,webrieyoutlinethemathematicalbasisoftheDEAmodels.
InSec-tion4,wedeveloptheproposedDEAmodelinthepresenceofintervaldataandundesirableoutputs.
InSection5,wedemon-stratetheapplicabilityoftheproposedmethodandexhibittheefcacyoftheprocedureusingasix-yearstudyof17combinedcyclepowerplantsinIran.
InSection6,wediscussthemanagerialimplicationsandinSection7,wepresentourconclusionandfutureresearchdirections.
2.
LiteraturereviewDEAisanon-parametricmethodforevaluatingtherelativeef-ciencyofDMUswithmultipleinputsandmultipleoutputs(Charnes,Cooper,&Rhodes,1978).
TherstDEAmodel(i.
e.
,theCCRmodel)wasproposedbyCharnesetal.
(1978)byconsideringtheconstantRSassumption.
Banker,Charnes,andCooper(1984)extendedtheCharnesetal.
's(1978)modelbyproposingtheBCCmodelandconsideringtheVRSassumption(Cooper,Seiford,&Tone,2007).
2.
1.
UncertaintyinDEAmodelsIngeneral,classicalDEAproblemsaresolvedundertheassump-tionthatthevaluesofparametersarespeciedpreciselyinacrispenvironment.
However,theobservedvaluesoftheinputandout-putdatainreal-worldproblemsareoftenimpreciseorvague.
Impreciseevaluationsisprimarilytheresultofunquantiable,incompleteandnon-obtainableinformation.
ThemostcommonformofuncertaintyinDEAproblemsoccurswhensomeoralloftherelevantdataarenotknownprecisely.
Thistypeofuncertaintyiscalledambiguity(Inuiguchi&Ramk,2000).
Theambiguityindatacanbemodeledusingthepossibilityapproachparameterizedthroughfuzzysets,ortheprobabilityapproachparameterizedthroughrandomvariables,orintervaldata.
Severaloptimizationmethodssuchasstochasticprogramming,fuzzymathematicalpro-gramming,andintervalmathematicalprogrammingareproposedtotakeintoaccountvariousuncertaintiesintheoptimizationpro-cess(Liu,Huang,Liu,Fuller,&Zeng,2003).
Stochasticprogrammingmethodsmodeluncertaintieswithprobabilitydistributionsderivedfromhistoricaldata(Peidro,K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773761Mula,Poler,&Lario,2009),whichmaynotbeavailableinmanyreal-lifeproblems.
Fuzzysettheory(Zadeh,1978)andpossibilitytheoryprovideaframeworktodealwithuncertaintiesintheformofvaguenessandambiguity(Dubois-Ferriere,Grossglauser,&Vetterli,2003).
Fuzzymathematicalprogrammingusuallyrequiresassumptionsaboutmembershipfunctionsandroughestimationmayincreasevaguenessandambiguity.
Intervalprogrammingisoftenusedwhentheavailabledataareinsufcientforaccuratelyestimatingdistributionfunctionsormembershipfunctions.
DespotisandSmirlis(2002)proposedamethodtodealwithimprecisedatainDEAmodels.
Theyproposedappropriatevariableexchangestoachievethesolutionstothelinearprograms.
Saati,Memariani,andJahanshahloo(2002)proposedafuzzyversionoftheCCRmodelrstproposedbyCharnesetal.
(1978)forrankingtheDMUswithasymmetricaltriangularfuzzynumbers.
TheytransformedthefuzzyCCRmodelintoacrisplinearprogrammingproblembyapplyinganalternativealpha-cutapproach.
Wang,Greatbanks,andYang(2005)proposedamethodformeasuringthelowerandupper-boundsofefciencyscoresofDMUs.
Thismethodiscapableofincorporatingdecisionmakers'preferencesfortheinputandoutputweights.
Kao(2006)appliedatwo-levelnon-linearmathematicalprogrammingtechniquetoderivethelower-andupper-boundsofefciencyscoresofDMUinthepres-enceofimpreciseness.
Emrouznejad,Rostamy-Malkhalifeh,Hatami-Marbini,andTavana(2012)proposedtwonewDEAmod-elsforevaluatingtherelativeefcienciesofDMUswithintervalinputandoutputdata.
Hatami-Marbini,Emrouznejad,andTavana(2011)hasclassiedthefuzzyDEAapproachesintofourclassesofthetoleranceapproach,thea-levelbasedapproach,thefuzzyrankingapproachandthepossibilityapproach.
Othertypesofuncertaintiesoftenobservedinreal-lifeprob-lemsincludeprobability,robustness,andintervaldata.
Li(1998)proposedastochasticDEAmodelbyassumingrandomdisturbancestorepresentthevariationsintheinput–outputdatastructure.
Li(1998)denedthestochasticefciencymeasureoftheDMUsusingjointprobabilisticcomparisonsoftheinputsandoutputswiththeotherDMUs.
Li(1998)proposedachance-con-strainedprogrammingproblemtosolvethestochasticDEAmodelandderivedthedeterministicequivalentsforthecasesofmultivar-iatesymmetricrandomdisturbancesandasinglerandomfactorintheproductionrelationships.
LahdelmaandSalminen(2006)introducedaDEAmethodtohandleuncertainorimprecisedataintheformofstochasticef-ciencymeasurescalledthestochasticmulti-criteriaacceptabilityanalysisDEAmethod.
KaoandLiu(2009)usedasimulationmethodtoobtaintheefciencydistributionoftheDMUSinDEA.
KaoandLiu(2009)usedtheconventionalmethodofaveragedatatorepresentstochasticvariables.
Thisresultedinefciencyscoreswhichweredifferentfromthemeanefcienciesofthedistribu-tionsestimatedfromthesimulationmethod.
Theyalsoappliedtheinterval-dataapproachandconcludedthattheintervalsweretoowidetoprovidevaluableinformation.
Finally,theyshowedthat,inthepresenceofmultipleobservationsforeachDMU,thestochastic-dataapproachproducedmorereliableandinformativeresultsthantheaverage-dataandinterval-dataapproaches.
Tavana,Shiraz,Hatami-Marbini,Agrell,andKhalil(2012)proposedthreefuzzyDEAmodelswithrespecttoprobability–possibility,probability–necessityandprobability–credibilitycon-straints.
Theyalsoconsideredtheprobabilityconstraintsandpre-sentedaverycomprehensivecasestudyforamilitarybaserealignmentandclosuredecisionprocessattheU.
S.
DepartmentofDefensetoillustratethefeaturesandtheapplicabilityofthepro-posedmodels.
SadjadiandOmrani(2008)proposedaDEAmethodbasedonrobustoptimizationtodealwithimprecisedataintheIranianelectricitydistributioncompanies.
SadjadiandOmrani(2008)proposedanewDEAmethodwithuncertainoutputparametersbasedontherobustoptimizationapproaches.
TheycomparedtheresultswithstochasticfrontieranalysisusingdatafromagroupofelectricitydistributioncompaniesandshowedtheeffectsofthedatauncertaintiesontheperformanceofDEAoutputs.
TheresultsindicatedthattherobustDEAapproachwasarelativelymorereliablemethodforefciencyestimationandtherankingofstrategies.
Sadjadi,Omrani,Abdollahzadeh,Alinaghian,andMohammadi(2011),Sadjadi,Omrani,Makui,andShahanaghi(2011)developedanewmethodwhichincorporatedtherobustcounterpartofsuper-efciencyDEA.
Theperturbationanduncertaintyindatawasassumedasanellipsoidalsetandtherobustsuper-efciencyDEAmodelwasextended.
Inthismethod,atargetsettingwasimple-mentedwiththeuncertaindataandthedecisionmakercouldsearchtheenvelopfrontierandndthetargetsbasedonhis/herpreferences.
Inordertosearchtheenvelopfrontier,theycombinedDEAwithamulti-objectivelinearprogrammingmethod.
Thecombinedmethodwascapableofhandlinguncertaintyinthedataandndingthetargetvaluesaccordingtothedecisionmakers'preferences.
Theresultsindicatedthattheircombinedmodelwassuitablefortargetsettingandforcasesofuncertaindata.
AbtahiandKhalili-Damghani(2011)proposedamathematicalformulationformeasuringtheperformanceofagilityinsupplychainsusingasingle-stagefuzzyDEA.
Khalili-Damghani,Taghavifard,Olfat,andFeizi(2011)appliedtheproposedformulationofAbtahiandKhalili-Damghani(2011)tomeasuretheefciencyofagilityinsupplychainsandusedsimulationtoranktheintervalefciencyscores.
Khalili-DamghaniandAbtahi(2011)measuredtheefciencyofjustintimeproductionssystemsusingafuzzyDEAapproach.
Khalili-DamghaniandTaghavifard(2012)proposedafuzzytwo-stageDEAapproachforperformancemeasurementinsupplychains.
Khalili-Damghani,Taghavifard,Olfat,andFeizi(2012)usedordinaldatainanewtwo-stageDEAapproachforagilityperfor-manceandillustratedtheefcacyoftheirapproachinasupplychain.
Khalili-DamghaniandTaghavifard(2013)performedsensi-tivityandstabilityanalysisintwo-stageDEAmodelswithfuzzydata.
TheyproposedseveralmodelsforcalculatingthestabilityradiusinDEAproblemswithconsiderableinputandoutputvariationsanduncertainties.
Khalili-DamghaniandTavana(2013)proposedanewnetworkDEAmodelformeasuringtheperfor-manceofagilityinsupplychains.
Theuncertaintyoftheinputandoutputdataweremodeledwithlinguistictermsandthepro-posedmodelwasusedtomeasuretheperformanceofagilityinareal-lifecasestudyofthedairyindustry.
2.
2.
Undesirableinputs/outputsinDEAmodelsADMUisusuallycalledefcientifitcanproducemaximumoutputsusingminimuminputs.
Inthiscase,theassociatedDMUissituatedontheefcientfrontieranditsefciencyscorewillbeequaltounity(Bankeretal.
,1984;Charnesetal.
,1978).
Occasion-ally,aDMUmaybeefcientifthevalueofsomeoutputsareaslowaspossible.
Thesetypesofoutputsarecalledundesirableandsev-eralstudieshavebeenproposedtomodelsuchoutputs(Chung,Fre,&Grosskopf,1997;Fre,Grosskopf,Lovell,&Pasurka,1989;Jahanshahloo,Lot,Shoja,Tohidi,&Razavyan,2005;Pathomsiri,Haghani,Dresner,&Windle,2008;Seiford&Zhu,2002).
Pathomsirietal.
(2008)assessedtheproductivityofUSairports,andthejointproductionofbothdesirableandundesirableoutput,i.
e.
delaysofight.
Thereareseveralmethodscandealwithunde-sirablefactorsinefciencyevaluation.
Oneofthemistotreattheundesirableoutputsasinputs(Wang,Wei,&Zhang,2012).
Anothermethodistouseanon-linearprogrammingapproachtodealwithundesirableoutputsandutilizethecomplementaryoftheundesir-ableoutputsinthestandardDEAmodel(Freetal.
,1989).
Other762K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773approachesincludemethodsthataredesignedtoincreasethedesirableoutputsandreducetheundesirableoutputsbysimulta-neousapplicationofthedirectionaldistancefunction(Chungetal.
,1997;Pathomsirietal.
,2008).
2.
3.
ReturntoscaleinDEAmodelsTheRSspecicallyseeksthemostproductivescalesizeforeachDMUintheproductionpossibilityset.
Bankeretal.
(1984)rstproposedtheconceptofthemostproductivescalesizeusingDEA.
Banker,Bardhan,andCooper(1996)discussedequivalenceandimplementationofalternativemethodsfordeterminingtheRSinDEA.
GolanyandYu(1997)proposedanewalgorithmtoesti-mateRSinDEAbasedonaslack-basedmethod.
2.
4.
ApplicationsofDEAmodelsinenergysystemsSueyoshiandGoto(2012a)proposedaDEAmethodanddis-criminantanalysistodeterminetheefciency-basedrankingofenergyrms.
TheproposedapproachwasappliedtoexaminetheperformanceoftheJapaneseelectricpowerindustry.
Theypro-posedtwotypesofoutputunication(i.
e.
,naturaldisposability,andmanagerialdisposability)forDEAenvironmentalassessmentbyusinganon-radialmodel.
SueyoshiandGoto(2012b)exploredhowtomeasureRSundernaturaldisposabilityandDSunderman-agerialdisposability.
Theyusedtheirmethodtocomparetheper-formanceofnationaloilrmswithinternationaloilcompanies.
SueyoshiandGoto(2012c)measureduniedefciencyundernaturalandmanagerialdisposabilityinradialDEAmodels.
TheyappliedtheirmethodtocomparetheperformanceofUScoal-redpowerplantsundertheIndependentSystemOperators/RegionalTransmissionOrganizationswithindependentpowerplants.
SueyoshiandGoto(2012d)reviewedthedisposabilityconceptsandreplacedtwotraditionaleconomicconceptsondisposabilitywithnaturalandmanagerialdisposability.
Theystudiedtheconceptualandmethodologicaldifferencesofweak/strongdispos-abilityandnatural/managerialdisposability,focusingupontheconceptofcongestionandtechnologicalinnovation.
TheyusedtheirmethodtocompareJapaneseelectricpowerrmswithman-ufacturingrms.
Theyshowedthatthemanufacturingrmsout-performedtheelectricpowerrmsundernaturaldisposability,whereastheoppositeresultedundermanagerialdisposability.
SueyoshiandGoto(2012e)suggestedabroaderfocusforDEAenvironmentalassessmentbymeasuringthemarginalrateoftransformationandtherateofsubstitutionbetweendesirableandundesirableoutputs.
TheyusedtheirmethodtoevaluatetheperformanceofUScoal-redpowerplantsandconcludedthattheregulationpolicyonNOxandSO2hadbeeneffectiveontheiremissioncontrolsundertheUSCleanAirAct.
TheyalsoshowedthattheregulationofCO2,amajorsourceoftheglobalwarmingandclimatechange,wasstillinsufcientintheUnitedStates.
SueyoshiandGoto(2012f)proposedamethodformeasuringRSundernaturaldisposabilityandDSundermanagerialdisposabilityusingDEA.
ThemeasurementofRSandDSwasformulatedbyincorporating''strongcomplementaryslacknessconditions''.
Mul-tiplereferencesetsandmultipleprojectionsintheRS/DSmeasure-mentwereproposed.
SueyoshiandGoto(2012f)showedthatsuchanalyticalcapabilitiesisessentialbuttheyhavenotbeenprevi-ouslyexploredinDEAenvironmentalassessmentforenergyindustries.
SueyoshiandGoto(2012g)proposedanon-radialDEAapproachformeasuringtheoperationalandenvironmentalperfor-manceofDMUs.
Environmentalperformancewascalculatedbasedontheundesirableoutputsoftheproductionprocesswhileopera-tionalperformancewasmeasuredbasedonthenormaloutputsofthesystem.
TheyappliedtheirmethodtoUScoalredpowerplantsandcomparedmethodologicalstrengthsanddrawbacksoftheradialandnon-radialmodelsusedforDEAenvironmentalassessments.
SueyoshiandGoto(2012h)proposedaDEAmethodbasedonradialandnon-radialprojection.
Theyconsideredapro-ductionprocesswhichproducednotonlydesirable(good)butalsoundesirable(bad)outputs.
Tounifythesetwotypesofoutputs,SueyoshiandGoto(2012h)usedtwotypesofdisposability,naturaldisposabilityandmanagerialdisposability.
TheydiscussedhowtomeasureRSundernaturaldisposabilityandDSundermanagerialdisposability.
TheyappliedtheirmethodtoUSfossilfuelpowerplantsandsuggestedapolicyimplicationforintroducingnewtechnologyforenvironmentalprotection.
Theyalsoarguedforthenecessityofdevelopingamethodologicalbasisforenergystudies.
SueyoshiandGoto(2013a)comparedfossilfuelpowerplantsinPennsylvania–NewJersey–MarylandandCaliforniawithrespecttooperationalandenvironmentalperformanceusingDEAmethodol-ogy.
Theyincorporatedstrategicconceptssuchasnaturalandmanagerialdisposabilityintothecomputationalprocessandpre-sentedamethodtomeasureRSundernaturaldisposabilityandtomeasureDSundermanagerialdisposability.
TheyshowedthatCaliforniaoutperformedPennsylvania–NewJersey–Marylandintermsofthethreeuniedefciencymeasures.
Theresultimpliedthatstrictregulationonundesirableoutputs,asfoundinCalifornia,wasimportantinenhancingtheperformanceofUSfossilfuelpowerplants.
SueyoshiandGoto(2013b)proposedaDEAmethodbyusingtheMalmquistindextoexaminethedegreeofafrontiershiftamongmultipleperiods.
Thefrontiershiftindicatedatechno-logicalprogressand/ormanagerialinnovationduringanobservedperiod.
Theyutilizedtheproposedapproachinanempiricalappli-cationandidentiedtherelationshipbetweenfuelmix,electricityandCO2amongtenindustrialnations.
Sueyoshi,Goto,andSugiyama(2013)proposedanewuseforwindowanalysisinDEAenvironmentalassessment.
Theyincorpo-ratedtheconceptofnaturalandmanagerialdisposabilityintothecomputationalframeworkofDEAandextendedthetwodispos-abilityconcepts.
TheyusedtheDEAwindowanalysisonadatasetfortheUScoal-redpowerplants.
Theirstudyshowedthatthecoal-redpowerplantshadgraduallyadheredtotheenviron-mentalprotectionsundertheCleanAirActandtheperformancewithrespecttomanagerialdisposabilityhadincreased.
ZhangandChoi(2013)proposedameta-frontiernon-radialMalmquistCO2emissionperformanceindex(MCPI)formeasuringdynamicchangesintotal-factorCO2emissionperformanceovertime.
Themeta-frontiernon-radialMCPImethodtookintoaccounttheincorporationofgroupheterogeneityandnon-radialslackintothepreviouslyintroducedMCPI.
ZhangandChoi(2013)examinedthedynamicchangesinCO2emissionperformanceanditsdecom-positionoffossilfuelpowerplantsinChina.
Theempiricalresultsrepresentedanincreaseintotal-factorCO2emissionperformanceasawholeandaU-shapedmeta-frontiernon-radialMCPIcurveforthesampleperiod.
Chen(2013)reexaminednon-additiveenvironmentalefciencymodelswithweakly-disposableundesirableoutputsthatwerepublishedintheenergyeconomicsliterature.
Hepresentedataxonomyofefciencymodelsfoundintheenergyeconomicsliteratureandillustratedsomelimitationsanddiscussedimplica-tionsofmonotonicityfromapracticalviewpoint.
Healsoformulatedavariablereturns-to-scaletechnologywithweakly-disposableundesirableoutputstoevaluatetheenergyefcienciesof23EuropeanUnionstates.
2.
5.
MotivationandcontributionThispaperismotivatedbytheneedtodevelopacomprehensiveperformanceevaluationmodelforcombinedcyclepowerplantsK.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773763wherethemodelcouldconcurrentlyconsideruncertainty(intervaldata)andundesirablefactors(outputs)anddetermine(1)therel-ativeefcienciesandinefcienciesofthepowerplants(DMUs);(2)themosteconomicscalesizefortheefcientDMUs;and(3)prac-ticalbenchmarksandreferencesetsforalltheinefcientDMUs.
Unfortunately,themodelsdevelopedintheliteraturehavespecicrestrictionsandlimitationsthatpreventedtheirapplicationtoourproblem.
Therefore,themodelproposedinthisstudywasdevel-opedtollthisresearchgap.
InordertodemonstrateourcontributiontotheliteratureinDEA,wecomparedthemethodproposedinthisstudywiththemethodsproposedin39papersfromtheliteraturebasedonthefollowingnineuniquefeaturesincluding:(1)considerationofuncertainty(i.
e.
,(1-a)fuzzy,(1-b)probabilistic/stochastic,(1-c)robust,and(1-d)interval);(2)considerationofundesirableout-puts(i.
e.
,(2-a)transferfunction,(2-b)treatmentofinputs,and(2-c)Inverseoutput);(3)determinationoftheefciencyscoresoftheDMUsasintervalvalues;(4)developmentofagroupofindicestodistinguishtheefcientandinefcientDMUs;(5)deter-minationofthemosteconomicscalesizefortheefcientDMUs;(6)determinationofpracticalbenchmarksfortheinefcientDMUs;(7)orientationofaprojectiontowardtheefcientfrontier(i.
e.
,(7-a)input,(7-b)output,and(7-c)mixed);(8)applicationintheeldofenergy;and(9)classicalDEAmodeling.
TheresultsarepresentedinTable1.
AsshowninTable1,themethodproposedinthisstudyhasseveraluniquefeatures.
Weuseintervaldatatomodeltheuncertainties,treattheundesirableoutputsasinputsduringthemathematicalmodelingphase,achieveintervalefciencyscoresfortheDMUs,provideclassicationtoranktheDMUs,ranktheDMUsusingamulti-attributedecisionmakingmethod,determinethemosteconomicscalesizefortheefcientDMUs,anddeterminetheprojectionandpracticalbenchmarksfortheinefcientDMUsbasedontheprojectiontowardtheefcientfrontier.
Although,thesefeatureshavebeenseparatelyreportedintheDEAliterature,tothebestofourknowledge,noonehasreportedasimultaneoustreatmentofthesefeaturesinonemodel.
Thecombinationoftheseuniqueandattractivefeaturesmakethemodelproposedinthisstudyrobustandapplicabletoreal-lifeperformancemeasurementproblems.
3.
BackgroundLetusconsidernj1;2;nDMUsunderevaluation.
EachDMUjisassumedtoproducesr1;2;soutputsrepresentedTable1Researchcontributions.
ResearchpaperResearchcontribution*(1)(2)(3)(4)(5)(6)(7)(8)(9)Aguirreetal.
(2011)UU(7-a)UUAzadeetal.
(2008)UU(7-a)UUBampatsouetal.
(2013)UU(7-a)UBankeretal.
(1984)UUU(7-a)UBianetal.
(2013)(2-b)U(7-a)UUCharnesetal.
(1978)UU(7-a)UChungetal.
(1997)(2-a)U(7-a)UDespotisandSmirlis(2002)(1-a)UU(7-a)Emrouznejadetal.
(2012)(1-d)UU(7-a)UFangetal.
(2013)UU(7-a)UUFreetal.
(1989)(2-c)U(7-a)UGolanyandYu(1997)UUU(7-a)UHatami-Marbinietal.
(2011)(1-a)UU(7-a),(7-b),(7-c)UKagawaetal.
(2013)UU(7-a)UUKao(2006)(1-a)UU(7-a)KaoandLiu(2009)(1-b),(1-d)UU(7-a)Khalili-DamghaniandTavana(2013)(1-a)(2-c)UUU(7-a)LahdelmaandSalminen(2006)(1-b)UU(7-a)Lee(2009)UUU(7-a)Li(1998)(1-b)UU(7-a)LiouandWu(2011)UUU(7-a)ULozano(2011)UUU(7-a)UMandal(2010)(2-b)UU(7-a)UPathomsirietal.
(2008)(2-a)U(7-a)UPuriandYadav(2013)(1-a)(2-b)UU(7-a)UPuriandYadav(2014)(1-a)(2-b)UU(7-a)URiccardietal.
(2012)(2-a)UU(7-a)UUSaatietal.
(2002)(1-a)UU(7-a)USadjadiandOmrani(2008)(1-c)UUU(7-a)USadjadi,Omrani,Abdollahzadehetal.
(2011),Sadjadi,Omrani,Makui,etal.
(2011b)(1-c)UUU(7-a)USalazar-Ordóezetal.
(2013)U(7-a)USueyoshiandGoto(2010)(2-b)UU(7-a)UTavanaetal.
(2012)(1-a),(1-b)UU(7-a)VazhayilandBalasubramanian(2013)(1-b)UU(7-a)UUWangetal.
(2005)(1-a)UU(7-a)Wuetal.
(2012)U(7-a)UWuetal.
(2013)(2-b)U(7-a)UZhangetal.
(2013)(2-a)U(7-a)UZhouetal.
(2013)UU(7-a)UMethodproposedinthisstudy(1-d)(2-b)UUUU(7-a)UU*Noteonresearchcontributions:(1)Considerationofuncertainty:(1-a)fuzzy,(1-b)probabilistic/stochastic,(1-c)robust,(1-d)interval.
(2)Considerationofundesirableoutputs:(2-a)transferfunction,(2-b)treatmentofinputs,(2-c)Inverseoutput.
(3)DeterminationoftheefciencyscoresoftheDMUsasintervalvalues.
(4)DevelopmentofgroupofindicestodistinguishbetweentheefcientandinefcientDMUs.
(5)DeterminationofthemosteconomicscalesizefortheefcientDMUs.
(6)DeterminationofpracticalbenchmarksfortheinefcientDMUs.
(7)Orientationoftheprojectiontowardtheefcientfrontier:(7-a)input,(7-b)output,(7-c)mixed.
(8)Applicationintheeldofenergy.
(9)ClassicalDEAmodeling.
764K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773byYjy1j;y2j;ymjandtoconsumemi1;2;minputsrepresentedbyXjx1j;x2j;xmj.
AllinputsandoutputsforalltheDMUsarenon-negativeandeachDMUhasatleastonestrictlypositiveinputandoutput.
TheCCRproductionpossibilitysetpro-posedbyCharnesetal.
(1978)isestimatedby(1).
TCCRX;YjXPXnj1kjXj;Y6Xnj1kjYjP0;kjP0;j1;.
.
.
;n()1TheBCCproductionpossibilitysetproposedbyBankeretal.
(1984)isestimatedby(2).
TBCCX;YjXPXnj1kjXj;Y6Xnj1kjYjP0;(Xnj1kj1;kjP0;j1;n)2TheCCRandBCCefciencyscorescanbeobtainedbyusingtheenvelopmentinput-orientedModels(3)and(4),respectivelywherexioandyrorepresenttheithinputandtherthoutputvectorofDMUounderevaluationinbothModels(3)and(4).
ADMUiscalledCCR-efcientifitsobjectivevalueinModel(3)isequaltounity.
minhCCRs:t:Xnj1kjxij6hCCRxio;i1;mXnj1kjyrjPyro;r1;skjP0;j1;n:3minhBCCs:t:Xnj1kjxij6hBCCxio;i1;mXnj1kjyrjPyro;r1;sXnj1kj1kjP0;j1;n:4ADMUiscalledBCC-efcientifitsobjectivevalueinModel(4)isequaltounity.
4.
ProposedDEAmodelsconsideringintervaldataandundesirableoutputInthissection,DEAmodelsinthepresenceofintervaldataandundesirableoutputsareproposedtomeasuretheRS.
ThemethodproposedbyGuoandWu(2013)iscustomizedtohandleundesir-ableoutputs.
Moreover,themethodproposedbyDespotisandSmirlis(2002)isadaptedtohandleintervalinputsandoutputs.
4.
1.
EfciencyscoresinthepresenceofintervaldataandundesirableoutputConsidernDMUswhichconsumeminputstoproducesout-puts.
LetusfurtherassumethatxijrepresentstheleveloftheithinputforDMUj;ydrjrepresentstheleveloftherthdesirableoutputforDMUj;andyur0jrepresentstheleveloftherthundesirableoutputforDMUjTheundesirableoutputsshouldbedecreasedtoimprovetheperformanceofaDMU.
Theuncertaintyoftheinputsandoutputsareconsideredbythepositiveintervaldataofxij2xLij;xUijhi,ydrj2yLdrj;yUdrjhi,andyur0j2yLur0j;yUur0jhi.
Moreover,theundesirableoutputsareconsideredasinputs.
Model(5)isproposedinthepresenceofintervaldataandundesirableoutputs.
minhCCRs:t:Xnj1kjxLij;xUijhi6hCCRxLio;xUio;i1;mXnj1kjyLdrj;yUdrjhiPyLdro;yUdro;r1;sXnj1kjyLur0j;yUur0jhi6hCCRyLur0o;yUur0o;r01;s0kjP0;j1;2;n5Model(5)cannotbesolvedinitscurrentform.
Weusedtheoptimisticandpessimisticcasesinordertocalculatetheupperandlower-boundsoftheefciencyscoreforeachDMU(Despotis&Smirlis,2002;Seiford&Zhu2002;Wangetal.
,2005).
Wecon-siderthepessimisticscenariowheretheDMUunderevaluationissettoitsworstsituationandalltheremainingDMUsaresettotheirbestsituationsandproposeModel(6)tocalculatethelower-boundoftheefciencyscoreforeachDMUinthepresenceofintervaldataandundesirableoutputs.
minhCCRls:t:Xnj1j–okjxLijkoxUio6hCCRlxUio;i1;2;mXnj1j–okjyUdrjkoyLdroPyLdro;r1;2;SXnj1j–okjyLur0jkoyUur0o6hCCRlyUur0o;r01;2;S0kjP0;j1;2;n:6Next,weconsidertheoptimisticscenariowheretheDMUunderevaluationissettoitsbestsituationandalltheremainingDMUsaresettotheirworstsituationsandproposeModel(7)tocalculatetheupper-boundoftheefciencyscoreforeachDMUinthepres-enceofintervaldataandundesirableoutputs.
minhCCRus:t:Xnj1j–olkjxUijkoxLio6hCCRuxLio;i1;2;mXnj1j–olkjyLdrjkoyUdroPyUdro;r1;2;SXnj1j–olkjyUur0jkoyLur0o6hCCRuyUur0o;r01;2;S0kjP0;8j1;2;n:7Solvingmodels(6)and(7)forallDMUswillresultintheintervalefciencyscoreofhCCRl;hCCRuhiforeachDMU.
K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–7737654.
1.
1.
PropertiesoftheproposedmodelsTheorem#1.
Model(6)isalwaysfeasibleandbounded.
Itsoptimalobjectivefunctionisequaltounity.
Proof.
ConsiderthefollowingsolutionforModel(6):kj0;j1;2;n;j1oko1hCCRl1ItisobviousthatthesolutiontoModel(6)isalwaysfeasible.
Therefore,independentofinputsandoutputsvalues,therealwaysexistsatleastonefeasiblesolutionforModel(6).
SincetheabovesolutionisfeasibleandtheobjectivefunctionofModel(6)isminimization,theoptimumvalueoftheobjectivefunctioninModel(6)isdenitelylessthanorequaltounityi:e:;hCCRl61:Consequently,wecanconcludethatModel(6)isalwaysbounded.
Thiscompletestheproof.
hTheorem#2.
Model(7)isalwaysfeasibleandbounded.
Itsoptimalobjectivefunctionisequaltounity.
Proof.
ConsiderasolutionforModel(7)asfollows:kj0;j1;2;n;j–oko1hCCRu1ItisobviousthatthesolutiontoModel(7)isalwaysfeasible.
Therefore,independentofinputsandoutputsvalues,therealwaysexistsatleastonefeasiblesolutionforModel(7).
SincetheabovesolutionisfeasibleandtheobjectivefunctionofModel(7)ismin-imization,theoptimumvalueoftheobjectivefunctioninModel(7)isdenitelylessthanorequaltouniti:e;hCCRu61:Conse-quently,itcanbeconcludedthattheModel(7)isalwaysbounded.
Thiscompletestheproof.
hCorollary#1.
TheBCCmodelsconstructedbyaddingPnj1kj1toModels(6)and(7)arealsoalwaysfeasibleandbounded.
Corollary#2.
TheBCC–CCRmodelsconstructedbyaddingPnj1kjP1toModels(6)and(7)arealsoalwaysfeasibleandbounded.
Corollary#3.
TheCCR–BCCmodelsconstructedbyaddingPnj1kj61toModels(6)and(7)arealsoalwaysfeasibleandbounded.
4.
2.
DeterminingtheRSinthepresenceofintervaldataandundesirableoutputsInordertodeterminetheRSforeachDMUinthepresenceofintervaldataandundesirableoutputs,theefciencyscoresarecalculatedbyconsideringtheassumptionsofVRS,DecreasingReturntoScale(DRS),andIncreasingReturntoScale(IRS).
FortheVRSsituation,theconstraintPnj1kj1isaddedtoModels(6)and(7)tocalculatethelower-boundandupper-boundoftheefciencyscoresfortheDMUsrepresentedbytheintervalhBCCl;hBCCu.
FortheDRSsituation,theconstraintPnj1kj61isaddedtoModels(6)and(7)tocalculatethelower-boundandupper-boundoftheefciencyscoresfortheDMUsrepresentedbytheintervalhCCRBCCl;hCCRBCCuhi.
FortheIRSsituation,thecon-straintPnj1kjP1isaddedtoModels(6)and(7)tocalculatethelower-boundandupper-boundoftheefciencyscoresfortheDMUsrepresentedbytheintervalhBCCCCRl;hBCCCCRuhi.
Astheprocedureisclear,theassociatedmodelsarenotpresentedhereforthesakeofbrevity.
Usingtheaforementionedprocedure,fourefciencyscoresarecalculatedforeachDMUfortheoptimisticscenarioandfourefciencyscoresarecalculatedforeachDMUforthepessimisticscenario.
Fig.
1depictstheefcientfrontier(consideringoneinputandoneoutput)fortheCCR,BCC,BCC–CCR,andCCR–BCCmodels.
Itisnotablethatthesefrontiersareestimatedforboththeoptimis-ticandpessimisticscenarios,respectively.
WethenproposethefollowingproceduretodeterminetheRSforeachDMU.
First,weclassifyeachDMUbasedonitsassociatedintervalefciencyscoreasfollows:ifthelowerandtheupper-boundsoftheDMUarebothequaltounity,theDMUisclassiedasE++;ifthelower-boundoftheefciencyscorefortheDMUislessthanunityandtheupper-boundoftheefciencyscorefortheDMUisequaltounity,theDMUisclassiedasE+;andifthelowerandtheupper-boundsoftheefciencyscoresoftheDMUarebothlessthanunity,theDMUisclassiedasE.
FortheDMUsclassiedintheE++group,thecross-efciencymethodisusedtomakeafullrankingandsequentiallydeterminetheRS(seeTable2).
Table1isalsousedtodeterminetheRSfortheDMUsclassiedintheE+group.
ThecomparisonsintheE+grouparesimplyaccom-plishedbyusingthelower-boundoftheefciencyscoregiveninTable2.
FortheDMUsclassiedintheEgroup,thefollowingstepsareproposed:Step1.
Wecalculatetherangeandthemeanvalueoftheinter-valefciencyscoreforeachDMUfortheCCR,BCC,BCC–CCR,andCCR–BCCmodelsusingboththeoptimisticandthepessimisticscenarios,respectively.
DRSIRSCRSA5A4A3A2CCR-BCCBCC-CCRA1BCCOutputInputFig.
1.
AreasfordifferentRSassumptions.
Table2EfciencyscorerelationsusedindeterminingtheRS.
DRSCRSIRSBCC–CCR>CCR–BCCBCC=CCRBCC–CCRKhalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773Step2.
WeusethecalculatedrangesandmeansandruntheTOPSISalgorithmproposedbyHwangandYoon(1981)andShih,Shyur,andLee(2007)forall4modelsbasedonboththeoptimisticandthepessimisticscenariostoassignarankingscoretoeachDMU.
TheTOPSISalgorithmisbasedonthedistancesbetweentheDMUsandtwodummyDMUs(i.
e.
,theidealDMUandtheanti-idealDMU).
TheDMUwhichisfarfromtheanti-idealDMUandneartheideal-DMU,simultaneously,isthebestchoice.
TheDMUwhichisfarfromtheidealDMUandneartheanti-idealDMU,simultaneously,istheworstchoice.
TheidealDMUandtheanti-idealDMUareidentiedbyusing(8)and(9),respectively.
AfXAMaxMeanvalue;YAMinRangeg8AfXAMinMeanvalue;YAMaxRangeg9ThedistancebetweeneachDMUandtheidealDMUandthedistancebetweeneachDMUandtheanti-idealDMUiscalculatedusing(10)and(11),respectively.
djAMeanjXA2RangejYA2qj1;2;n10djAMeanjXA2RangejYA2qj1;2;n11Eq.
(12)isusednexttocalculateaclosenesscoefcient(CC)indexforeachDMUunderconsiderationCCjdjAdjAdjA8j1;2;n12Finally,theDMUsarerankedbasedonthedecreasingorderoftheirCCs.
Asaresult,fourprimaryranks,associatedwiththeCCR,BCC,CCR–BCC,andBCC–CCR,areassignedtoeachDMU.
Table1canbeusednexttodeterminetheRSbasedontheserankings.
4.
3.
FinalrankingoftheDMUsAfterdeterminingtheRS,decisionmakersmaybeinterestedinassigningauniquerankingtoeachDMU(consideringthefourdifferentrankings).
WecustomizethemethodproposedbySoleimani-DamanehandZarepisheh(2009)(basedonShannon'sentropy)todetermineanaluniquerankingforeachDMUinpresenceoftheintervaldata,undesirableoutputsandseveralRSassumptions.
ThedecisionmakingmatrixcontainstheordinalvaluesoftheDMUs'rankingsachievedbasedontheproposedTOPSISmethoddescribedintheprevioussection.
WehadassumedthatnDMUshavebeenevaluatedbyaDEAmodelunderkcriteria(k=1,2,K).
TheefciencyresultsarelistedinthematrixEnk.
EachrowoftheEmatrixcorrespondstoaDMUandeachcolumncorrespondstoaranking.
ThefollowingstepsareproposedtoassignauniquerankforeachDMU:Step1.
WeuseEq.
(13)andnormalizetherankingsinthedecisionmatrix.
RjkRjkPnj1Rjk;j1;2;n;k1;2;K13whereRjkistherankingorderofDMUjinRSk.
Step2.
WeuseEq.
(14)andcomputetheentropyvalue(ek).
eke0Xnj1EjklnEjk;k1;2;K14wheree0istheentropyconstantandisequaltoe0=(lnn)1Step3.
Wesetdk=1ekasthedegreeofdiversicationfork=1,2,K.
Step4.
WethenuseEq.
(15)tonormalizedk.
Fig.
2.
Combinedcyclepowerplant.
K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773767wkdkPKk1dk;k1;2;K15wherewkrepresentstherelativeimportanceoftherankingachievedbytheRSoftypek.
Step5.
WenallyuseEq.
(16)andcalculatethefollowingef-ciencyindex.
bjXKk1wkEjk;j1;2;n16wherebjisthenalrankingofDMUj,andEjkistheefciencyscoreofDMUjconsideringtheRSoftypek.
AllDMUsarerankedbasedonadecreasingorderofbj.
5.
CasestudyandresultsAcombinedpowerplantworkswithgasandsteamturbines.
Thegasturbineusesnaturalgasorgasolinefuelstogenerateelectricityandthesteamturbineusesthewasteheatfromthegasturbinetogenerateelectricity.
Theprocessisveryefcientsinceexhaustheat(thatwouldotherwisebelostthroughtheexhauststack)isre-usedthroughoutthesystem.
Agasturbinecompressestheairandmixesitwiththefuel.
Thefuelisburnedandtheresultanthotairexpandsandspinstheturbineblades.
Thespinningturbinebladesdrivesageneratorandthegeneratorconvertsthespinningmotionintoelectricitypower.
Theexhaustheatgeneratedinthegasturbineissenttoaheatrecoverysteamgenerator.
Thisgeneratorusesthegasturbineexhaustheattoproducesteamanddeliversittothesteamturbine.
Thesteamproducesadditionalenergyinthesteamturbineandthesteamtur-binedeliversthisenergytothegeneratordriveshaft.
Thegeneratorconvertsthisenergyintoelectricity.
TheprocessofgeneratingelectricityinacombinedcyclepowerplantisbrieyshowninFig.
2.
ThemethodproposedinthisstudywasusedbytheIranianPowerGeneration,Transmission,andDistributionManagementCompany(TAVANIR)toevaluate17combinedcyclepowerplantsinIranduringasixyearperiod.
TAVANIRisresponsibleforelectric-itygeneration,transmissionanddistributioninIran.
Sixvariableswerechosenastheinputsandoutputstoestimatetheefciencyscoresofthe17combinedcyclepowerplantsunits.
FossilfuelistheprimaryinputinthecombinedcyclepowerplantsmanagedbyTAVANIR.
ThedesirableoutputistheunitsofproducedenergyandtheundesirableoutputsaregasessuchasCO2,SO2,SO3,andNOx.
Eachpowerplantconsumessomecombinationoffuelstopro-duceunitsofelectricityenergy,emissions,andpollutions.
Fig.
3representstheschematicviewofaDMUasapowerplant.
Themeasurementunitusedforthedesirableoutputsisthousandkilo-wattsperhourandthemeasurementunitusedfortheinputsislitersorm3.
Themeasurementunitusedfortheunde-sirableoutputsiston.
Table3presentstheinputandoutputdatausedinthisstudy.
Consideringtheoptimisticandpessimisticscenarios,fourintervalefciencyscoreswerecalculatedbasedondifferentRSassumptionsforeachcombinedcyclepowerplant(i.
e.
,CCR,BCC,CCR–BCC,andBCC–CCR).
Theintervalefciencyscoresarepre-sentedinTable4.
Thecombinedcyclepowerplants(DMUs)areclassiedaccord-ingtothelowerandupper-boundsoftheirefciencyscorestodeterminetheRS.
GiventhatalltheDMUsinallthemodelshaveanupper-boundefciencyscoreof1andalower-boundefciencyscorelessthan1,weuseTable1toclassifyalltheDMUsintheE+group.
TheRSvaluesassociatedwiththeDMUsarealsosumma-rizedinTable4.
Fig.
4representstheefciencyscoresoftheDMUsforalltheRSassumptionsunderbothoptimisticandpessimisticscenarios.
TheproposedTOPSISmethodwasusedtofullyranktheDMUs.
ThecomputationalresultsoftheTOPSISmethodarepresentedinTable5foralltheRSassumptions.
ThenalrankingoftheDMUsCO2(Y2j)SO2(Y3j)SO3(Y4j)Fossilfuels(GasandGasoline)ElectricityInputsDesirableandUndesirableCombinedCyclepowerplantNOx(Y5j)DMUFig.
3.
ApowerplantasaDMU.
Table3Intervalinputsandoutputsofthepowerplantsduringthesix-yearstudy.
DMUInputDesirableoutputUndesirableoutput(GAS/Ton)Fuel(M3)Electricitypower(1000KW/Hr.
)NoxSO2CO2SO3LowerboundUpperboundLowerboundUpperboundLowerboundUpperboundLowerboundUpperboundLowerboundUpperboundLowerboundUpperbound11,002,2431,534,3814,663,8205,948,1233.
651.
86.
62338301500.
12971,5091,298,1124,821,2965,657,3923.
74.
42.
14.
72367272700.
131,331,4571,831,0987,220,8517,699,5125.
35.
82.
86.
73478363100.
14766,6581,117,3223,781,8434,628,5202.
83.
71.
74.
81779225000.
1524,2131,060,942356,9633,184,6310.
63.
20.
42.
231821190061,045,4551,283,5415,339,7805,975,6863.
84.
41.
3325452806007412,442758,1421,925,8562,631,2101.
72.
30.
11.
110521557008446,0941,017,3391,836,7934,289,0041.
83.
61.
14.
21089222900.
191,244,5201,820,7374,222,7967,935,5714.
380.
212.
52806478800.
2101,056,1821,410,6805,126,2566,213,1383.
44.
40.
23226228020011311,239635,2571,820,2092,106,0151.
61.
913.
3979109100.
112204796,6055152,128,41002.
6051159500.
1131,234,9222,303,4684,500,1699,886,1025.
28.
34.
111.
4320949930.
10.
214422,191905,8741,770,3322,761,5531.
92.
90.
42.
4122218480015147,6832,769,6345,008,7721,030,0085.
58.
539.
23546553500.
116161,614928,6371,258,5702,678,9961.
94.
51.
47.
7985266100.
1171,298,6881,961,3144,785,7535,898,7175.
35.
70.
543382382800.
1768K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773werethencalculatedusingShannon'sentropymethod.
Thepri-maryrankingofTOPSISmethodandthenalrankingmeasureofShannon'sentropymethodarepresentedinTable5.
AsshowninTable6,thebestDMUisDMUNo.
3andtheworstDMUisDMUNo.
16.
AsshowninTable3,somepowerplantsshouldbeexpanded(i.
e.
,thoseundertheIRSscenario)sincetheyhaveachievedtheirmostproductivescalesize.
Someotherpowerplantsshouldbecondensed(i.
e.
,thoseundertheDRSscenario)sothattheycanachievetheirmostproductivescalesize.
ThereferencesetsoftheCCR,BCC,BCC–CCR,andCCR–BCCmodelsaresummarizedinTable7.
Consideringreturntoscaleassumptions,andusingthelinearcombinationoftheinputsandoutputsofthereferencesetofaDMU,theDMUcanbeprojectedtowardstheefcientfrontier(i.
e.
,mostproductivescalesize).
Itshouldbenotedthatsincetheproposedmodelsinthispaperareinput-oriented,theprojectionisaccomplishedbasedontheinputvalues.
Forexample,DMU6canbeprojectedtowardstheefcientfron-tierusingitsreferencesetprovidedinTable7.
GiventhefactthatTable4EfciencyresultsfordifferentDEAmodels.
DMU*ClassBCCCCR-BCCCCRBCC-CCRRSofDMULowerboundUpperboundLowerboundUpperboundLowerboundUpperboundLowerboundUpperbound1E+0.
32306810.
00072710.
00072710.
3230681IRS2E+0.
39169010.
00083110.
00083110.
3916901IRS3E+0.
58118010.
00093410.
00093410.
5811801IRS4E+0.
27295610.
00079010.
00079010.
2729561IRS5E+0.
00047210.
00047210.
00007910.
0000791DRS6E+0.
47366510.
00089410.
00089410.
4736651IRS7E+0.
00064210.
00064210.
00058110.
0005811DRS8E+0.
00044910.
00044910.
00038710.
0003871DRS9E+0.
16407710.
00041410.
00041410.
1640771IRS10E+0.
44024710.
00086010.
00086010.
4402471IRS11E+0.
00091710.
00091710.
00078410.
0007841DRS12E+0.
23076910.
23076910.
00003710.
0000371DRS13E+0.
18321810.
00042310.
00042310.
1832181IRS14E+0.
00054110.
00054110.
00045010.
0004501DRS15E+0.
22158310.
00042510.
00042510.
2215831IRS16E+0.
00037610.
00037610.
00022210.
0002221DRS17E+0.
29826710.
00058710.
00058710.
2982671IRS*Thenameofthepowerplantsareintentionallyomittedtoprotecttheiranonymity.
(a)BCCModel(b)CCRModel(c)CCR-BCCModel(d)BCC-CCRModel00.
20.
40.
60.
811.
21234567891011121314151617PessimiscOpmisc00.
20.
40.
60.
811.
21234567891011121314151617PessimiscOpmisc00.
20.
40.
60.
811.
21234567891011121314151617PessimiscOpmisc00.
20.
40.
60.
811.
21234567891011121314151617PessimiscOpmiscFig.
4.
OptimisticandpessimisticefciencyscoresoftheDMUs.
K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773769DMU6isclassiedasanIRSDMU(checktheefciencyscoresofDMU6inTable3andusethecategoriesinTable1),itcanbepro-jectedtowardstheefcientfrontierusingalinearcombinationofDMU5andDMU15(i.
e.
,themembersofreferencesetforDMU6intheBCC–CCRcolumn).
Sincetheinputsandoutputsoftherefer-encesetarealsointervalvalues,apracticalsolutionisrequiredtoprojecttheDMUstowardtheefcientfrontier.
ThisprojectionshouldreducetheinputsandundesirableoutputsforDMU6basedonitsintervalefciencyscoreandthecombinationoftheassoci-ateddecisionvariablesofthereferencesetforDMU6.
ThefollowingstrategiesareproposedtoprojecttheinefcientDMUstowardstheefcientfrontierinthepresenceofundesirableoutputsandintervaldata:OptimisticScenario:Thisscenarioconsidersarisk-seekingdecisionmaker.
Therefore,weassumethattheupper-boundoftheefciencyscoreisachievedandtheinputsandoutputshaveachievedtheirbestlevels.
Inotherwords,theDMUunderassess-mentcanachievetheupper-boundoftheefciencyscorewhileconsumingthelower-boundoftheinputsandproducingtheupper-boundofthedesirableoutputsandthelower-boundoftheundesirableoutputs.
PessimisticScenario:Thisscenarioconsidersarisk-aversedecisionmaker.
Therefore,weassumethatthelower-boundofef-ciencyscoreisachievedandtheinputsandoutputshaveachievedtheirworstlevels.
Inotherwords,theDMUunderassessmentcanachievethelower-boundoftheefciencyscorewhileconsumingtheupper-boundoftheinputsandproducingthelower-boundofthedesirableoutputsandtheupper-boundofundesirableoutputs.
Table8presentstheparametersofagivenDMUundertheoptimis-ticandpessimisticscenarios.
Forexample,DMU6hasthefollowingparametersundertheoptimisticscenario:X16=1,045,455,Y16=5,975,686,Y1'6=3.
8,Y2'6=1.
3,Y3'6=2545,Y4'6=0.
TheRSforDMU6isclassiedintheIRSgroupbasedontheresultsprovidedinTable3.
Intheoptimisticscenario,theef-ciencyscoreofDMU6is1basedontheresultsprovidedinTable4.
Table5TOPSISresultsfortheprimaryrankings.
DMUBCCCCR–BCCCCRBCC–CCRdj(A+)dj(A)CCjdj(A+)dj(A)CCjdj(A+)dj(A)CCjdj(A+)dj(A)CCj10.
288570.
360780.
555590.
257190.
000390.
001520.
000230.
000770.
768600.
2885780.
361150.
5558520.
211850.
437500.
673740.
257070.
000500.
001970.
000110.
000880.
884410.
2118570.
437880.
67393300.
6493510.
256960.
000620.
0024200.
00100100.
64973140.
344600.
304750.
469310.
257120.
000460.
001790.
000160.
000840.
838750.
3446050.
305130.
4696250.
649250.
000100.
000160.
257480.
000100.
000410.
000954.
6E050.
046630.
64964.
7E057.
2E0560.
120200.
529150.
814880.
257000.
000570.
002254.
5E050.
000950.
955120.
120200.
529530.
8149970.
649060.
000290.
000450.
257290.
000290.
001150.
000390.
000600.
606240.
649120.
000600.
0009380.
649278.
1E050.
000120.
257508.
1E050.
000310.
000610.
000390.
390000.
649340.
000390.
0006090.
466330.
183020.
281850.
257544.
3E050.
000160.
000580.
000420.
420330.
466330.
183400.
2822100.
157560.
491790.
757340.
257040.
000540.
00218.
3E050.
000910.
916630.
157560.
492160.
75748110.
648750.
000600.
000930.
256980.
000600.
002340.
000160.
000830.
832240.
648900.
000830.
00128120.
391770.
257580.
3966800.
2575810.
00100000.
6497300130.
444930.
204420.
314800.
257535.
3E050.
000200.
000570.
000430.
430460.
444930.
204800.
31520140.
649170.
000180.
000280.
257400.
000150.
000710.
000540.
000460.
460150.
649270.
000460.
00071150.
402040.
247310.
380860.
257535.
52E00.
000210.
000560.
000430.
43230.
402040.
247690.
38122160.
64935000.
25758000.
000790.
000200.
206120.
649530.
000200.
00031170.
316300.
333050.
512890.
257350.
000230.
000910.
000380.
000610.
613210.
316300.
333430.
51317Table6TOPSIS(primaryrankings)andShannon'sentropy(nalrankings).
770K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773ThereferencesetforDMU6areDMU5andDMU15.
Theassociateddecisionvariablesarek5=0.
6971170andk15=0.
3028830.
Intheoptimisticscenario,allotherDMUs,includingtherefer-encesetofDMU6(i.
e.
,DMU5andDMU15),areassumedtobeintheirworstsituations.
Wehave:X15=1,060,942,Y15=356,963,Y1'5=3.
2,Y2'5=2.
2,Y3'5=2129,Y4'5=0.
X1,15=2,769,634,Y1,15=5,008,772,Y1',15=8.
5,Y2',15=9.
2,Y3',15=5535,Y4',15=0.
1.
TheprojectionofDMU6ontheefcientfrontieriscalculatedasalinearcombinationoftheinputsandtheoutputsofthereferencesetusingk5=0.
6971170andk15=0.
3028830asfollows:X16=61609.
96,Y16=5,339,780,Y1'6=2.
084127,Y2'6=1.
187496,Y3'6=1295.
706,Y4'6=0.
Asshownhere,thispracticalbenchmark,intheoptimisticscenario,consumesfewerinputsandproducesfewerundesirableoutputsincomparisonwithDMU6.
TheremainingDMUscanbedescribedsimilarlybasedontheoptimisticorthepessimisticviewpoints.
6.
ManagerialimplicationsInthispaper,wedevelopedaperformanceassessmentsystembasedonDEAandusedtheproposedsystemtomeasuretheper-formanceofcombinedcyclepowerplants(DMUs)withintervaldataandundesirableoutputs.
Wedemonstratedtheapplicabilityoftheproposedmethodandexhibitedtheefcacyoftheprocedureusingintervaldataandundesirableoutputswithasix-yearstudyof17combinedcyclepowerplantsinIran.
Apowerplantcon-sumesfossilfuelsandproduceselectricityasadesirableoutputandpollutinggasesasundesirableoutputs.
Itisimperativetoimprovetheprocessofefciencymeasurementandeconomicscalesizeincombinedcyclepowerplantssinceagoodestimateofemissionsandpollutionsproducedbythepowerplantcanresultinpracticalstrategiesforsustainabledevelopment.
7.
ConclusionandfutureresearchdirectionsAgoodestimateofemissionsandpollutionsproducedbycombinedcyclepowerplantscanhelpdeveloppracticalstrategiesforfurthersustainableprogress.
Performanceassessmentofanelectricitypowerplantanddeterminingitseconomicscaleischallengingandcomplex.
Thisassessmentusuallyrequiresacare-fulconsiderationofmultipleandoftenconictingfactors.
Theper-formanceassessmentprocessoftenbecomesmorecomplicatedbecause:(1)thedataassociatedwithsomeofthesefactorsarerep-resentedwithintervalvalues;and(2)thesepowerplantsroutinelyproduceemissionswhichareconsideredundesirableoutputs.
InthispaperwedevelopedaperformanceassessmentsystembasedonDEAandusedtheproposedsystemtomeasuretheper-formanceofcombinedcyclepowerplants(DMUs)withintervaldataandundesirableoutputs.
Weshowedthattheproposedapproachisabletohandleundesirableoutputs.
Moreover,theuncertaintyofdataduringwasmodeledusingintervaldata.
Sev-eralRSassumptionsweretestedtodeterminethebesteconomicsizeforacombinedcyclepowerplantinthepresenceofintervaldataandundesirableoutputs.
Theproposedmethodsuggestsprac-ticalrecommendationsforre-sizingthecombinedcyclepowerplantsandimprovingtheiroverallefciency.
Wepresentedacomprehensiveperformanceevaluationframe-workforcombinedcyclepowerplants.
Weneededtoconcurrentlyconsiderintervaldataandundesirableoutputsanddetermine:therelativeefcienciesandinefcienciesofthepowerplants,themosteconomicscalesizefortheefcientpowerplants,andpracticalbenchmarksandreferencesetsforallinefcientpowerplants.
Unfortunately,themodelsdevelopedintheliteraturecouldnotbeusedtosatisfythespecicrequirementsinthisproblems.
Con-sequently,wedevelopedtheperformanceevaluationframeworkproposedinthisstudyto:(1)modeltheuncertaintiesintheinputTable8ParametersofaDMUundertheoptimisticandpessimisticscenarios.
CCRCCR–BCCBCCBCC–CCROptimisticScenariohCCRhCCRLhCCRBCChCCRBCCLhBCChBCCLhBCCCCRhBCCCCRLxijxLij;8i;jxijxLij;8i;jxijxLij;8i;jxijxLij;8i;jydrjyUdrj;8r;jydrjyUdrj;8r;jydrjyUdrj;8r;jydrjyUdrj;8r;jyurjyLdrj;8r;jyurjyLdrj;8r;jyurjyLdrj;8r;jyurjyLdrj;8r;jPessimisticScenariohCCRhCCRUhCCRBCChCCRBCCUhBCChBCCUhBCCCCRhBCCCCRUxijxUij;8i;jxijxUij;8i;jxijxUij;8i;jxijxUij;8i;jydrjyLdrj;8r;jydrjyLdrj;8r;jydrjyLdrj;8r;jydrjyLdrj;8r;jyurjyUdrj;8r;jyurjyUdrj;8r;jyurjyUdrj;8r;jyurjyUdrj;8r;jTable7DMUreferencesetsanddecisionvariables.
DMUReferencesetsDecisionvariable(kj)BCC–CCRCCRCCR–BCCBCCBCC–CCRCCRCCR–BCCBCC113,5121213,5k5=0.
77920,k13=0.
2207k12=2.
1912k12=2.
1912k5=0.
7792,k13=0.
2207213,5121213,5k5=0.
7557,k13=0.
2442k12=2.
2652k12=2.
2652k5=0.
7557,k13=0.
2442315,13,5121215,13,5k5=0.
3989,k13=0.
5811,k15=0.
1988E01k12=3.
3926k12=3.
3926k5=0.
3989,k13=0.
5811,k15=0.
1988E01413,5121213,5k5=0.
9108,k13=0.
8911E01k12=1.
7768k12=1.
7768k5=0.
9108,k13=0.
8911E01512121212k12=0.
1677k12=0.
1677k12=1k12=1615,5121215,5k5=0.
6971,k15=0.
3028k12=2.
5082k12=2.
5088k5=0.
6971,k15=0.
3028712121212k12=0.
9048k12=0.
9048k12=1k12=1812121212k12=0.
8629k12=0.
8629k12=1k12=1913,5121213,5k5=0.
8450,k13=0.
15491k12=1.
9840k12=1.
9840k5=0.
8450,k13=0.
15491015,5121215,5k5=0.
7271,k15=0.
2728k12=2.
4084k12=2.
40849k5=0.
7271,k15=0.
27281112121212k12=0.
8551,k12=0.
8551k12=1k12=1125555k5=0.
1617E03k5=0.
1617E03k5=1k5=11315,5121215,5k5=0.
8151,k15=0.
1848k12=2.
114k12=2.
1143k5=0.
8151,k15=0.
18481412121212k12=0.
8317k12=0.
8317k12=k12=11513,9,5121213,9,5k5=0.
7070,k9=0.
7139E01,k13=0.
2215k12=2.
3532k12=2.
3532k5=0.
7070,k9=0.
7139E01,k13=0.
22151612121212k12=0.
5913k12=0.
5913k12=1k12=11715,13,5121215,13,5k5=0.
7653,k13=0.
1662,k15=0.
6845E01k12=2.
2485k12=2.
2485k5=0.
7653,k13=0.
1662,k15=0.
6845E01K.
Khalili-Damghanietal.
/ExpertSystemswithApplications42(2015)760–773771andoutputdatathroughintervaldata;(2)considerundesirableoutputs;(3)determinetheefciencyscoresoftheDMUsasinter-valvalues;(4)developagroupofindicestodistinguishtheef-cientandinefcientDMUs;(5)determinethemosteconomicscalesizefortheefcientDMUs;and(6)determinepracticalbenchmarksfortheinefcientDMUs.
Wedemonstratedtheapplicabilityoftheproposedmethodandexhibitedtheefcacyoftheprocedureusingintervaldataandundesirableoutputswithasix-yearstudyof17combinedcyclepowerplantsinIran.
Apowerplant,representedasaDMU,con-sumesfossilfuelsandproduceselectricityasadesirableoutputandpollutinggasesasundesirableoutputs.
Theefciencyscoreswerecalculatedforeachpowerplantduringtheplanningperiod.
Theeconomicscalesize,assessment,ranking,andpracticalsugges-tionforimprovementweresuggestedforeachpowerplant.
Theproposedapproachhadpromisingresultsanditsimplementationwasstraightforward.
Thebenetsofourmodelmightbestillnas-centbutthepotentialscouldbesignicant.
Themethodproposedinthisstudylenditselftomanyreal-lifeperformancemeasure-mentproblemswithintervaldataandundesirableoutputs.
Weencourageresearcherstousetheapproachproposedinthisstudyinotherapplications(e.
g.
,banking,supplychainnetworkperformancemeasurementandre-scaling,healthcaresystemsevaluation,exiblemanufacturingsystemassessment,transporta-tionsystemevaluation,etc.
).
Wemodeledtheuncertaintiesintheinputandoutputdatawithintervaldatainthisstudy.
Aninterest-ingstreamofresearchistomodeltheseuncertaintieswithfuzzydata,probabilisticdata,and/orgreydata.
Finally,wedeterminedanintervalefciencyscoreforeachDMUandidentiedtheef-cientandinefcientDMUsbasedontheseintervalefciencyscores.
Anotherinterestingstreamofresearchistoconductsensi-tivityanalysisonthestabilityoftheseintervalefciencyscoresortestthereliabilityofgroupingDMUsasefcientorinefcient.
AcknowledgementTheauthorswouldliketothanktheanonymousreviewersandtheeditorfortheirinsightfulcommentsandsuggestions.
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